Vacuum gauges generally need to be calibrated against pressure. Traditionally, vacuum gauges were used in vacuum chambers in which the pressure can easily be varied, but a new application for vacuum gauges is to integrate them on electronic chips or micro-electro-mechanical systems (MEMS) which are enclosed in hermetic vacuum packages in order to monitor pressure. In this case pressure cannot be controlled, and calibration of vacuum gauges enclosed within vacuum packages (die-level or wafer-level) is challenging. One approach is to calibrate the vacuum gauge against pressure prior to packaging under vacuum, but the packaging process generally requires annealing of the device, which may change the properties of the materials of the vacuum gauge, thus rendering the calibration invalid. Calibration against pressure after packaging requires a destructive test, opening the package to vary the vacuum level. It is possible to calibrate a subset of gauges from a batch and use this calibration for all the other gauges in the batch, but variations in material properties make this approach inaccurate.
An alternative to calibration is to model the response of the gauge as a function of pressure, and to fit this model to the response of the gauge during operation in order to extract the pressure. This requires accurate knowledge of the material properties and dimensions of the gauge in order to predict its response. This may not be possible, especially when using micro-fabrication techniques, where within-wafer and wafer-to-wafer variations are common.
Thermal conductivity gauges are a well-known type of vacuum gauge and have been commercially available for several decades. More recently, micro-fabrication technology has been applied to the fabrication of miniaturized gauges, and there are several patents disclosing design and fabrication of this type of device: U.S. Pat. Nos. 4,682,503, 5,347,869, 5,557,972, 8,171,801 B2, 8,449,177 B2, 9,335,231 B2.
A large body of work has studied ways of improving design and operation of thermal conductivity gauges. Given that these gauges were traditionally used in vacuum chambers where pressure could easily be varied, there is little work on calibration methods. The most widely studied topic is compensation of the effect of ambient temperature variations, for example U.S. Pat. Nos. 6,023,979, 6,474,172, 6,658,941 B1, 7,249,516, 7,331,237 B2.                U.S. Pat. No. 8,504,313 proposes a method of calibration where calibration against pressure is replaced by calibration of temperature. However, this method still requires exposure to atmospheric pressure, so is not adapted to micro-fabricated thermal conductivity vacuum gauges packaged inside closed dies.        U.S. Pat. No.7,385,199 B2 discloses a method for using an infrared bolometer pixel as a thermal-conductivity vacuum gauge. A detailed calibration strategy is also described, but again requires calibration against pressure.        U.S. Pat. No. 8,171,801 B2 presents a micro-fabricated thermistor gauge and an operation method, whereby the voltage response of the gauge is measured at two different temperatures. This method is designed to suppress the effect of certain parameters, such as ambient temperature variations, but requires the voltage difference between the two temperatures to be calibrated against pressure.        
A large number of scientific articles have been published on the topic of micro-fabricated thermal conductivity gauges. Several articles present gauges which are packaged inside sealed dies, but the gauges are either calibrated before packaging, or a hole is made in the package to measure its pressure response. These articles include:                J. Mitchell, G. R. Lahiji, and K. Najafi, “An Improved Performance Poly-Si Pirani Vacuum Gauge Using Heat-Distributing Structural Supports,” J. Microelectromech. Syst., vol. 17, no. 1, pp. 93-102, February 2008.        Junseok Chae, J. M. Giachino, and K. Najafi, “Fabrication and Characterization of a Wafer-Level MEMS Vacuum Package With Vertical Feedthroughs,” J. Microelectromech. Syst., vol. 17, no. 1, pp. 193-200, February 2008.        L. Zhang, B. Jiao, S. Yun, Y. Kong, C. Ku, and D.-P. Chen, “A CMOS Compatible MEMS Pirani Vacuum Gauge with Monocrystal Silicon,” Chinese Phys. Lett., vol. 34, no. 2, p. 25101, 2017.        R. Kuljic et al., “Microelectromechanical system-based vacuum gauge for measuring pressure and outgassing rates in miniaturized vacuum microelectronic devices,” J. Vac. Sci. Technol. B, vol. 29, no. 2, p. 02B114, 2011.        
Several articles have published useful models of the response of the gauges:                F. Völklein, M. Grau, A. Meier, G. Hemer, L. Breuer, and P. Woias, “Optimized MEMS Pirani sensor with increased pressure measurement sensitivity in the fine and high vacuum regime,” J. Vac. Sci. Technol. A, vol. 31, no. 6, p. 61604, 2013.        C. H. Mastrangelo and R. S. Muller, “Microfabricated thermal absolute-pressure sensor with on-chip digital front-end processor,” IEEE J. Solid-State Circuits, vol. 26, no. 12, pp. 1998-2007, 1991.        P. Eriksson, J. Y. Andersson, and G. Stemme, “Thermal characterization of surface-micromachined silicon nitride membranes for thermal infrared detectors,” J. Microelectromech. Syst., vol. 6, no. 1, pp. 55-61, March 1997.        
In summary, many patents and articles have published on the subject of thermal conductivity gauges, and micro-fabricated gauges in particular. The topics of design, fabrication, and operation are well covered, but to our knowledge no work has been published which presents a calibration method which does not require calibration against pressure.